Characteristic cycles and the microlocal geometry of the Gauss map, II

We show that any Weyl group orbit of weights for the Tannakian group of semisimple holonomic 𝒟 {{\mathscr{D}}} -modules on an abelian variety is realized by a Lagrangian cycle on the cotangent bundle. As applications we discuss a weak solution to the Schottky problem in genus five, an obstruction for the existence of summands of subvarieties on abelian varieties, and a criterion for the simplicity of the arising Lie algebras.

Symmetry and Relative Equilibria of a Bicycle System

In this paper, we study the symmetry of a bicycle moving on a flat, level ground. Applying the Gibbs – Appell equations to the bicycle dynamics, we previously observed that the coefficients of these equations appeared to depend on the lean and steer angles only, and in one such equation, a term quadratic in the rear wheel’s angular velocity and a pseudoforce term would always vanish. These properties indeed arise from the symmetry of the bicycle system. From the point of view of the geometric mechanics, the bicycle’s configuration space is a trivial principal fiber bundle whose structure group plays the role of a symmetry group to keep the Lagrangian and constraint distribution invariant. We analyze the dimension relationship between the space of admissible velocities and the tangent space to the group orbit, and then employ the reduced nonholonomic Lagrange – d’Alembert equations to directly prove the previously observed properties of the bicycle dynamics. We then point out that the Gibbs – Appell equations give the local representative of the reduced dynamic system on the reduced constraint space, whose relative equilibria are related to the bicycle’s uniform upright straight or circular motion. Under the full rank condition of a Jacobian matrix, these relative equilibria are not isolated, but form several families of one-parameter solutions. Finally, we prove that these relative equilibria are Lyapunov (but not asymptotically) stable under certain conditions. However, an isolated asymptotically stable equilibrium may be achieved by restricting the system to an invariant manifold, which is the level set of the reduced constrained energy.

A quantum solution for efficient use of symmetries in the simulation of many-body systems

A many-body Hamiltonian can be block-diagonalized by expressing it in terms of symmetry-adapted basis states. Finding the group orbit representatives of these basis states and their corresponding symmetries is currently a memory/computational bottleneck on classical computers during exact diagonalization. We apply Grover’s search in the form of a minimization procedure to solve this problem. Our quantum solution provides an exponential reduction in memory, and a quadratic speedup in time over classical methods. We discuss explicitly the full circuit implementation of Grover minimization as applied to this problem, finding that the oracle only scales as polylog in the size of the group, which acts as the search space. Further, we design an error mitigation scheme that, with no additional qubits, reduces the impact of bit-flip errors on the computation, with the magnitude of mitigation directly correlated with the error rate, improving the utility of the algorithm in the Noisy Intermediate Scale Quantum era.

Nonlinear mode interactions in a counter-rotating split-cylinder flow

The flow in a split cylinder with each half in exact counter rotation is studied numerically. The exact counter rotation, quantified by a Reynolds number $\mathit{Re}$ based on the rotation rate and radius, imparts the system with an $O(2)$ symmetry (invariance to azimuthal rotations as well as to an involution consisting of a reflection about the mid-plane composed with a reflection about any meridional plane). The $O(2)$ symmetric basic state is dominated by a shear layer at the mid-plane separating the two counter-rotating bodies of fluid, created by the opposite-signed vortex lines emanating from the two endwalls being bent to meet at the split in the sidewall. With the exact counter rotation, the additional involution symmetry allows for steady non-axisymmetric states, that exist as a group orbit. Different members of the group simply correspond to different azimuthal orientations of the same flow structure. Steady states with azimuthal wavenumber $m$ (the value of $m$ depending on the cylinder aspect ratio $\unicode[STIX]{x1D6E4}$) are the primary modes of instability as $\mathit{Re}$ and $\unicode[STIX]{x1D6E4}$ are varied. Mode competition between different steady states ensues, and further bifurcations lead to a variety of different time-dependent states, including rotating waves, direction-reversing waves, as well as a number of slow–fast pulse waves with a variety of spatio-temporal symmetries. Further from the primary instabilities, the competition between the vortex lines from each half-cylinder settles on either a $m=2$ steady state or a limit cycle state with a half-period-flip spatio-temporal symmetry. By computing in symmetric subspaces as well as in the full space, we are able to unravel many details of the dynamics involved.

ON GENERALIZATION OF SPECIAL FUNCTIONS RELATED TO WEYL GROUPS

Weyl group orbit functions are defined in the context of Weyl groups of simple Lie algebras. They are multivariable complex functions possessing remarkable properties such as (anti)invariance with respect to the corresponding Weyl group, continuous and discrete orthogonality. A crucial tool in their definition are so-called sign homomorphisms, which coincide with one-dimensional irreducible representations. In this work we generalize the definition of orbit functions using characters of irreducible representations of higher dimensions. We describe their properties and give examples for Weyl groups of rank 2 and 3.

ON CUBATURE RULES ASSOCIATED TO WEYL GROUP ORBIT FUNCTIONS

The aim of this article is to describe several cubature formulas related to the Weyl group orbit functions, i.e. to the special cases of the Jacobi polynomials associated to root systems. The diagram containing the relations among the special functions associated to the Weyl group orbit functions is presented and the link between the Weyl group orbit functions and the Jacobi polynomials is explicitly derived in full generality. The four cubature rules corresponding to these polynomials are summarized for all simple Lie algebras and their properties simultaneously tested on model functions. The Clenshaw-Curtis method is used to obtain additional formulas connected with the simple Lie algebra <em>C</em>₂.

Faces of Platonic solids in all dimensions

This paper considers Platonic solids/polytopes in the real Euclidean space {\bb R}^n of dimension 3 ≤n< ∞. The Platonic solids/polytopes are described together with their faces of dimensions 0 ≤d≤n− 1. Dual pairs of Platonic polytopes are considered in parallel. The underlying finite Coxeter groups are those of simple Lie algebras of typesAn,Bn,Cn,F4, also called the Weyl groups or, equivalently, crystallographic Coxeter groups, and of non-crystallographic Coxeter groupsH3,H4. The method consists of recursively decorating the appropriate Coxeter–Dynkin diagram. Each recursion step provides the essential information about faces of a specific dimension. If, at each recursion step, all of the faces are in the same Coxeter group orbit,i.e.are identical, the solid is called Platonic. The main result of the paper is found in Theorem 2.1 and Propositions 3.1 and 3.2.